Genus 1 Curves in Severi--Brauer Surfaces

TL;DR

This paper explores the relationship between genus one curves in Severi-Brauer surfaces and properties of the underlying algebra, revealing a surprising connection involving Galois structures and skew commuting pairs.

Contribution

It provides a novel description linking genus one curves in Severi-Brauer surfaces to algebraic properties of the division algebra, especially through Galois cohomology and torsion point structures.

Findings

Connection between Galois structure of E[3] and skew commuting pairs in D⊗F K

Characterization of which elliptic curves E arise in this setting

Description of which genus one curves C arise via Galois cohomology

Abstract

In a talk at the Banff International Research Station in 2015 Asher Auel asked questions about genus one curves in Severi-Brauer varieties $SB(A)$. More specifically he asked about the smooth cubic curves in Severi-Brauer surfaces, that is in $SB(D)$ where $D/F$ is a degree three division algebra. Even more specifically, he asked about the Jacobian, $E$, of these curves. In this paper we give a version of an answer to both these questions, describing the surprising connection between these curves and properties of the algebra $A$. Let $F$ contain $\rho$, a primitive third root of one. Since $D/F$ is cyclic, it is generated over $F$ by $x,y$ such that $xy = \rho{yx}$ and we call $x,y$ a skew commuting pairs. The connection mentioned above is between the Galois structure of the three torsion points $E[3]$ and the Galois structure of skew commuting pairs in extensions $D \otimes_F K$. Given a description of which $E$ arise, we then describe, via Galois cohomology, which $C$ arise.